Since many aspects of the bibliography style are idiosyncratic (and chosen by publishers) it's not clear that such descriptions exist. Some of the major styles are described by specific publications such as the Chicago Manual of Style (the so-called Chicago style) or by academic societies/associations such as the American Psychological Association (the so-called APA style) or the Modern Language Association (the so-called MLA style.) Probably every major academic association or society (ACM, AMS, IEEE, APS, etc.) has its own version of a bibliography style. But even within these major types of styles, different publishers or journals may have their own variants of the styles.
One good source of general descriptions of these styles (and of powerful LaTeX methods of implementing them) is the
biblatex package. Its documentation provides descriptions of the major types bibliography and citation styles. This may be the closest thing you can get to descriptions in words, other than looking at sources of specific styles associated to particular academic societies.
Roughly speaking, there are five main types (with a lot of variations)
- Numeric (citations of the form [1,3] etc.; variations include whether the bibliography items appear in citation order or in alphabetical order)
- Author-Year (citations of the form Author (year))
- Alpha (like Author-Year, but citations are of the form [AuthorYY] or similar.)
- Author-Title (citations of the form Author, Title)
- Verbose (a full bibliography entry appears (usually in a footnote) upon first citation and some shorthand version is used for subsequent citations.) This style often dispenses with a separate bibliography altogether.
Especially with author-title and verbose schemes, and sometimes with author-year schemes, some form of ibid referencing may also be used: (roughly, recently cited references aren't repeated, but use ibid instead.)
Of course there are many variations in the ordering and punctuation of elements within the bibliography items themselves. Some styles are highly abbreviated, while others are not. I don't think there's a single resource that you can look at to find out about all of them.
In more practical terms, most of us work in fields where there are a few commonly used schemes, and we just need to know those. In my field, the Author-Year scheme is used, and in fact there is a move towards unification of the bibliography styles for all journals. Other fields may not be so enlightened. :-)